LETTER XIII.

OF THE EQUATION OF TIME; OR THE DIFFERENCE BETWEEN MEAN TIME AND APPARENT.

It is a circumstance worthy of observation, that, excepting the rotation of the earth upon its axis, there is no one body in nature, with which we are acquainted, whose motion is perfectly uniform and regular (m). The sun, in his apparent journey through the heavens, is supposed by the vulgar to furnish us with an accurate and just measure of time; but in this they are greatly mistaken. Astronomers have found that the motion of the sun is very unequal; and therefore equal time, which flows on for ever in the same manner, cannot be truly measured by the sun's motion. Mean, or equal time, is that which is shown by a perfectly well regulated clock or watch; and in order that the apparent time, as shown by a true sun-dial, may agree with this, it must be corrected by proper equations.

The difference between mean and apparent time depends upon two causes, the obliquity of the ecliptic with respect to the equator, and the unequal motion of the earth in an elliptical orbit. I shall first explain the effects of these causes separately considered, and then the united effects resulting from their combination. But, before we

(m) See the observation that has been made upon this part of the subject, in page 189.

proceed to these particulars, it will be proper to remind you, that whenever the motion of the sun is spoken of, it is not to be understood in a positive sense, as if he actually removed from one part of space to another, but only as an appearance occasioned by the real motion of the earth in a contrary direction. The phænomena are exactly the same; and astronomers sometimes mention one, and sometimes the other, according as they find it most convenient for their purpose.

This being premised, it may be observed, that since the earth's axis is perpendicular to the plane of the equator, any equal portions of this circle, 'will, by means of the earth's rotation upon its axis, pass over the meridian in equal times; and so, in like manner, would any equal portions of the ecliptic, provided it were parallel to, or coincident with the equator. But as this is not the case, the daily motion of the earth upon its axis will carry unequal portions of it over the meridian in equal times; the difference being always proportional to the obliquity and, as some parts of the ecliptic are much more obliquely situated with respect to the equator than others, those differences will be unequal amongst themselves.

Suppose, for example, that the sun and a star were to set out together from one of the equinoctial points, and to move continually through equal arcs in equal times; the star in the equator, and the sun in the ecliptic: then it is plain that the star, moving in the equator, would always return to the meridian exactly at the end of every twenty-four

hours, as measured by a well regulated clock, that keeps equal time; but the sun, moving in the ecliptic, would come to the meridian, sometimes sooner than the star, and sometimes later, according to their relative situations; and they would never be. found upon that circle exactly together, except on four days of the year; namely, on the 20th of March, when the sun enters Aries; on the 21st of June, when he enters Cancer; on 23d of September, when he enters Libra; and on the 21st of December, when he enters Capricorn.

But lest a verbal description should be found insufficient, I shall endeavour to make it more intelligible by means of a figure. For this purpose, let z ❤z~(Pl. vII. fig. 3.) be the earth; ZFRx its axis; abcd, &c. the equator; ABCD, &c. the northern half of the ecliptic, from to, on the side next to the eye; and MNOP the southern half, on the opposite side, from to v. In like manner, let A, B, C, D, &c. be the boundaries of equal portions of the ecliptic, gone through in equal times by the sun; and a, b, c, d, &c. equal portions of the equator, described in equal times by the star; also let zz be the meridian.

Then, as the sun moves obliquely in the ecliptic, and the star directly in the equator, a degree, or any number of degrees, between and F on the ecliptic, must be nearer to the meridian z vz, thạn a degree, or any corresponding number of degrees on the equator, from toƒ; and the more so as The sun, therefore, comes

they are more oblique.

to the meridian sooner every day, whilst he is in

the quadrant F, than the star does in the quadrant f; and, as the motion of the fictitious star in the equator, answers to the motion of a well regulated clock, it is plain that the solar noon, in this case, will precede the noon by the clock.

On the contrary, whilst the sun describes the second quadrant of the ecliptic FGHIKL, from to , he will come later to the meridian every day than the star, which moves through the second quadrant of the equator, from ƒ to; for the points G, H, I, K, L, being farther from the meridian than the corresponding points g, h, i, k, l, they must be later in coming to it; and as the sun and star arrive at the point at the same moment, they must then both come to the meridian together at the instant when it is noon by the clock.

Again, in departing from Libra, through the third quadrant, the sun going through MNOPQ towards, and the star through mnopq, towards r; the former will come to the meridian every day sooner than the latter, till the sun arrives at the point, and the star at the point r, and then they will both come to the meridian at the same time. And, in like manner, as the sun moves through the fourth quadrant STUvw, from towards , and the star through the quadrant stuvw, from r towards, the former will come later every day to the meridian than the latter, till they both arrive at the point, and then they will make it noon at the same time with the clock.

This part of the equation of time, may be made still more familiar by means of a globe; for if a

small black patch be put on every tenth or fifteenth degree, both of the equator and ecliptic, beginning at the point, and the globe be turned round slowly to the westward, you will observe that all the patches from Aries to Cancer, and from Libra to Capricorn, will come to the meridian sooner than their corresponding patches on the equator; and all those from Cancer to Libra, and from Capricorn to Aries, will come to the meridian later than their correponding patches on the equator: whilst the patches at the beginning of Aries, Cancer, Libra, and Capicorn, being on, or even with those on the equator, show that the sun and star, will either meet there, or are even with each other, and, for that reason, must come to the meridian at the same time.

Mr. Ferguson, whom I have chiefly followed in this article, proposes the following method for showing the difference between solar, sidereal, and equal time. Suppose two little balls are made to move equally round a celestial globe, by means of clock-work; one always keeping in the ecliptic, and gilt with gold, to represent the real sun; and the other, keeping always in the equator, and silvered, to represent a fictitious sun; and let it be so contrived, that whilst these two balls move once round the globe, according to the order of the signs, the globe shall be made to turn three hun. dred and sixty-six times round its axis, westward.

Then, as the motion of the globe is uniform, any fixed star will come to the meridian in equal times, and make in all three hundred and sixty-six revo